Is it true that for any sets A and B, P(A) (B)=P(A )? Justify you… - Vidyadip

Question

Is it true that for any sets A and $\text{B},\text{ P(A)}\cup\text{P(B)}=\text{P(A}\cup\text{B})$? Justify your answer.

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Answer

This is a false statement
Let, A = {1} and B = {2}
Then,
$\text{P(A)}=\{\phi,\{1\}\}$
and $\text{P(A)}=\{\phi,\{2\}\}$
$\therefore\text{ P(A)}\cup\text{P(B)}=\{\phi, \{1\}, \{2\}\}$
Now,
$\text{A}\cup\text{B}=\{1, 2\}$
and $\text{P(A}\cup\text{B})=\{\phi, \{1\}, \{2\}, \{1, 2\}\}$
Hence, $\text{P(A)}\cup\text{P(B)}\not=\text{P(A}\cup\text{B).}$

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